Kostant's indecomposability conjecture for projective functors in type A

Let $\theta_x$ and $\theta_y$ be projective functors acting on the principal block $4\mathcal{O}_04$ of the BGG category for $4\mathfrak{g}4$, and let $L_y$ be the simple module indexed by $y\rtimes W$. Indecomposability conjecture. If $4\mathfrak{g}\cong\mathfrak{sl}_n4$, then, for any $x,y\in W$, the module $\theta_x L_y$ is either indecomposable or zero. The conjecture concerns the structure of projective-functor images of simple modules and was proposed in the cited work; no resolution is supplied here.